Numerical modelling apparatus

ABSTRACT

A numerical modelling apparatus and method of performing numerical modelling are described. An input unit receives signals giving information relating to a set of assets. A processor unit is arranged to provide a Risk Relation Matrix v having elements that represent a relationship of risk related to a respective pair of the assets. Each element is a scalar product of two risk vectors, such that each of the assets has an associated risk vector according to the elements of the risk relation matrix. The Risk Relation Matrix v is decomposed into eigenvectors and eigenvalues according to V=E·Λ·E′, where E is a set of eigenvectors of the risk matrix v in columns, Λ is the corresponding diagonal eigenvalue matrix, and E′ is the transpose of E. Components of each of the risk vectors are derived in the basis of unit independent risks by the corresponding row of the matrix product E·Λ 1/2  relating to each of the assets. An output unit is arranged to output the components of each of the risk vectors as a risk vector data set onto a tangible computer-readable recording medium.

BACKGROUND

1. Technical Field

The present invention relates generally to the field ofcomputer-implemented apparatus for numerical modelling, and moreparticularly to computer apparatus and computer-implemented method fornumerical modelling of financial assets.

2. Description of Related Art

It is known to provide computer-implemented financial modelling toolsthat attempt to model or predict possible outcomes in relation tovarious financial assets. These tools are relatively complex, and aretypically implemented using computer apparatus with sufficient memory,processing power, etc to perform the necessary calculations underlyingthe relevant model.

There is a long-standing economic theory of equilibrium as the equalityof supply and demand in an exchange economy. This equilibrium theory hasbeen developed since the first general formulation by Walras in 1874.

In the related art, some asset pricing models based upon cashflows havebeen developed outside of an equilibrium framework, for instance byWilliams. Later methods like the Capital Asset Pricing Model (CAPM) ofSharpe, Lintner and Mossin are based upon the Mean-Variance PortfolioConstruction Approach of Markowitz and the Tobin Separation Theorem.

The Mean-Variance approach uses a set of expected returns (the Meanpart) and a variance-covariance matrix (the Variance part). ThisMean-Variance approach presupposes that a probability distribution forreturns is available and only the first two moments of the distributionare employed, so that higher order moments, like skew and kurtosis, areneglected. Thus the statistical basis of the Mean-Variance and itslimitations have been pointed out by many parties since it wasdeveloped.

The CAPM was built with the Mean-Variance framework as its descriptionof risk. The CAPM often described as an equilibrium theory because ifall market participants share the same views on expected return andexpected covariance of return then they will all hold the sameportfolio, suitably levered to allow for their risk tolerance, andmarket clearing requires this to be the market cap weighted portfolio ifmarkets are efficient.

The CAPM identifies the Market Portfolio as the efficient portfolio. TheCAPM lead directly to the development of Market Capitalizationinvestment products and indices that have grown to become the defactobenchmark in many markets, in particular developed equity markets. Thatis, it is known to provide computer-implemented financial models thatoutput indices representing relative performance of a portfolio offinancial assets, or provide portfolio constructions that allow aportfolio of the financial assets to be bought or sold.

As further background information see:

-   Williams, John Burr. 1938. The Theory of Investment Value, Harvard    University Press, Cambridge Mass.-   Lintner, John. 1965. The Valuation of Risk Assets and the Selection    of Risky Investments in Stock Portfolios and Capital Budgets. Review    of Economics and Statistics. 47:1, pp. 13-37.-   Markowitz, Harry. 1952. Portfolio Selection. Journal of Finance.    7:1, pp. 77-91.-   Markowitz, Harry. 1959. Portfolio Selection: Efficient    Diversification of Investments. Cowles Foundation Monograph No. 16.    New York: John Wiley & Sons, Inc.-   Markowitz, Harry. 2008. CAPM Investors Do Not Get Paid for Bearing    Risk: A Linear Relation Does Not Imply Payment for Risk. Journal of    Portfolio Management. 34:2 (Winter), pp. 91-94-   Mossin, January 1966, Equilibrium in a Capital Asset Market,    Econometrica, 34, pp. 768-783.-   Roll, Richard. 1977. A Critique of the Asset Pricing Theory's Tests.    Journal of Financial Economics, 4:2, pp. 129-176.-   Sharpe, William F. 1964. Capital Asset Prices: A Theory of Market    Equilibrium under Conditions of Risk. Journal of Finance. 19:3, pp.    425-442.-   Tobin, James. 1958. Liquidity Preference as Behavior Toward Risk.    Review of Economic Studies. 67, pp. 65-86.-   Choueifaty, Yves and Yves Coignard, 2008. Toward Maximum    Diversification, Journal of Portfolio Management. Fall 2008, 35:1,    pp. 40-51.

A problem arises in that the market capitalization weighted proxies tothe market portfolio have been demonstrated ex-post to be less efficientthan a number of other methods of portfolio construction over longperiods. Other criticisms have been made of the assumptions of the CAPMand ability to empirically prove the CAPM, such as in Roll. This haslead to a number of other approaches being developed to attempt toprovide more efficient portfolios than the market capitalizationweighted portfolios. These have included fundamental indexation byArnott in U.S. Pat. No. 7,620,577, non-constant functions ofcapitalization weights by Fernholz in U.S. Pat. No. 5,819,238, anddiversification based constructions by Choueifaty in US2008/222,252A1,and other more generic equally weighted and minimum variance portfolios.

It is now desired to provide an apparatus and method to implement afinancial modelling tool. In particular, an apparatus and method aredesired which output more detailed and/or more accurate financialmodelling information.

SUMMARY OF THE INVENTION

According to the present invention there is provided an apparatus andmethod as set forth in the appended claims. Other features of theinvention will be apparent from the dependent claims, and thedescription which follows.

In one aspect there is provided a numerical modelling apparatus, theapparatus comprising: an input unit arranged to receive signalscontaining data having information relating to a set of assets; aprocessor unit arranged to: a) provide a Risk Relation Matrix v having aplurality of elements, wherein each of the elements represents arelationship of risk related to a respective pair of the assets and eachelement is given by a scalar product of two risk vectors, such that eachof the assets has an associated risk vector according to the elements ofthe risk relation matrix; b) decompose the Risk Relation Matrix v intoeigenvectors and eigenvalues according to: V=E·Λ·E′ wherein E is a setof eigenvectors of the risk matrix v in columns, Λ is the correspondingdiagonal eigenvalue matrix, and E′ is the transpose of E; and c) derivecomponents of each of the risk vectors in the basis of unit independentrisks by the corresponding row of the matrix product E··^(1/2) relatingto each of the assets; and an output unit arranged to output thecomponents of each of the risk vectors as a risk vector data set onto atangible computer-readable recording medium.

In one aspect, the input unit is arranged to receive risk informationrelating to the set of assets and provide the risk information to theprocessor unit.

In one aspect, the input unit is arranged to receive the Risk RelationMatrix as the risk information and to provide the Risk Relation Matrixto the processor unit.

In one aspect, the input unit is arranged to receive the risk vectors asthe risk information and the processor unit is arranged to provide theRisk Relation Matrix from the risk vectors.

In one aspect, the processor unit is arranged to provide a dataset ofportfolio weights w for each of the assets according to:

w∝E·Λ^(−1/2)·1

where E is a set of eigenvectors of the risk matrix v in columns and Λis the diagonal eigenvalue matrix and 1 is a column vector all of whoseelements are equal to one.

In one aspect, the output unit is arranged to output the portfolioweightings w relating to the set of assets.

In one aspect, the processor unit is arranged to provide a dataset ofexpected returns R for each of the assets as a vector according to:

R∝E·Λ^(1/2)·1

where E is a set of eigenvectors of the risk matrix v in columns and Λis the corresponding diagonal eigenvalue matrix and 1 is a column vectorall of whose elements are equal to one.

In one aspect, the output unit is arranged to output the expectedreturns R relating to the set of assets.

In one aspect, the processor unit is arranged to obtain a dataset ofportfolio weightings w relating to each of the assets, wherein theportfolio weightings w are provided as a column vector from:

$D = \frac{w^{\prime} \cdot E \cdot \Lambda^{1/2} \cdot 1}{\sqrt{w^{\prime} \cdot V \cdot w}}$

where w represents a column vector containing portfolio weights, E isthe eigenvector matrix and Λ is a corresponding diagonal eigenvaluematrix of the Risk Relation Matrix V for the set of assets, and D is anObjective Function relating to the set of assets.

In one aspect, the output unit is arranged to output the value of theObjective Function as a Diversification Measure relating to the set ofassets.

In one aspect, the input unit is arranged to receive one or moreconstraints relating to the set of assets, and the processor unit isarranged to optimise the Objective Function D by variation of theportfolio weights subject to the constraints.

In one aspect, the processor unit is further arranged to collect theportfolio weights w at a plurality of time intervals.

In one aspect, the processor unit is further arranged to provide a PriceIndex based on the price information and the collected weights relatingto the set of assets.

In one aspect, the output unit is arranged to output the Price Index asa Benchmark Index relating to the set of assets.

In one aspect there is provided a method of performing numericalmodelling using a computer apparatus, the method comprising: receivingsignals at an input unit of the computer apparatus, the signalscontaining data having information relating to a set of assets;providing a Risk Relation Matrix v using a processor unit of thecomputer apparatus, wherein the Risk Relation Matrix v comprises aplurality of elements, wherein each of the elements represents arelationship of risk related to a respective pair of the assets and eachelement is given by a scalar product of two risk vectors, such that eachof the assets has an associated risk vector according to the elements ofthe risk relation matrix; decomposing the Risk Relation Matrix v intoeigenvectors and eigenvalues according to: V=E·Λ·E′ wherein E is a setof eigenvectors of the risk matrix v in columns, Λ is the correspondingdiagonal eigenvalue matrix, and E′ is the transpose of E; and derivingcomponents of each of the risk vectors in the basis of unit independentrisks by the corresponding row of the matrix product E·Λ^(1/2) relatingto each of the assets; and outputting the components of each of the riskvectors as a risk vector data set onto a tangible computer-readablerecording medium.

In one aspect there is provided a tangible computer-readable recordingmedium having recorded thereon instructions which when implemented by acomputer apparatus perform a method of numerical modelling, wherein themethod comprises the steps of: receiving signals at an input unit of thecomputer apparatus, the signals containing data having informationrelating to a set of assets; providing a Risk Relation Matrix v using aprocessor unit of the computer apparatus, wherein the Risk RelationMatrix v comprises a plurality of elements, wherein each of the elementsrepresents a relationship of risk related to a respective pair of theassets and each element is given by a scalar product of two riskvectors, such that each of the assets has an associated risk vectoraccording to the elements of the risk relation matrix; decomposing theRisk Relation Matrix v into eigenvectors and eigenvalues according to:V=E·Λ·E′ wherein E is a set of eigenvectors of the risk matrix v incolumns, Λ is the corresponding diagonal eigenvalue matrix, and E′ isthe transpose of E; and deriving components of each of the risk vectorsin the basis of unit independent risks by the corresponding row of thematrix product E·Λ^(1/2) relating to each of the assets; and outputtingthe components of each of the risk vectors as a risk vector data setonto a tangible computer-readable recording medium.

In one aspect, an apparatus is provided that receives a set of inputsrepresenting data relevant to a set of financial assets, makescalculations based on those inputs according to a predetermined model,and produces one or more outputs representing data relevant to the setof assets. In one aspect, the apparatus includes an input unit, aprocessor unit, and an output unit. In one aspect, the processor unit isarranged to receive signals containing data having information, such asprice information and/or risk information, relating to a set of assets;provide a Risk Relation Matrix having a plurality of elements, whereineach of the elements represents a relationship of risk related to arespective pair of the assets and each element is given by a scalarproduct of two risk vectors, such that each of the assets has anassociated risk vector according to the elements of the risk relationmatrix; derive components of each of the risk vectors in the basis ofunit independent risks; and output the components of each of the riskvectors as a risk vector data set.

In one aspect, a method is provided for performing numerical modellingusing a computer apparatus. The method may comprise receiving signalscontaining data having information relating to a set of assets;providing a Risk Relation Matrix having a plurality of elements, whereineach of the elements represents a relationship of risk related to arespective pair of the assets and each element is given by a scalarproduct of two risk vectors, such that each of the assets has anassociated risk vector according to the elements of the risk relationmatrix; deriving components of each of the risk vectors in the basis ofunit independent risks; and outputting the components of each of therisk vectors as a risk vector data set.

In one aspect, an equilibrium is established in the model whenindependent risk factors with the same magnitude of risk have identicalexpected excess returns. In one aspect the model performs calculationson the basis of a risk relation matrix. In one aspect, the modelprovides asset independent risk vectors.

In one aspect, a portfolio is determined with optimal expected excessreturn to risk given a risk magnitude for each asset and a riskcorrelation between each pair of assets. In one aspect, adiversification measure is provided for any portfolio of assets giventhe asset risks and the risk correlation between each pair of assets. Inone aspect there is provided a history of portfolio weights that form abenchmark index for any selected universe of assets.

In one aspect, a set of portfolio weightings is output. In one example,optimal constrained or unconstrained portfolio weights are output.Suitably, a portfolio of financial assets is constructed according tothe weightings. Advantageously, such a portfolio may have improvedperformance relative to the other portfolio constructions. In oneaspect, such improved performance is measured by an indicator including,for example, a risk adjusted return of the set of assets for assetprices, risks and expected returns. In one aspect, the portfolio mayhave the optimal expected excess returns for unit risk.

In one aspect, a diversification measure for portfolios is output. Thatis, the apparatus and method outputs a measure that can be used tocompare different portfolios.

In one aspect, a benchmark index is output that can be employed tomeasure performance, such as a benchmark index that is used by investorsto measure the performance of their investments.

BRIEF DESCRIPTION OF THE DRAWINGS

For a better understanding of the invention, and to show how exampleembodiments may be carried into effect, reference will now be made tothe accompanying drawings in which:

FIG. 1 is a schematic view of a numerical modelling apparatus accordingto one example embodiment;

FIG. 2 is a flowchart showing processes performed within the examplenumerical modelling apparatus.

DETAILED DESCRIPTION OF THE EXAMPLE EMBODIMENTS

The example embodiments will be described in relation to acomputer-implemented financial modelling tool that supports a model inrelation to various financial assets.

FIG. 1 is a schematic view of a numerical modelling apparatus accordingto one example embodiment. In FIG. 1, the apparatus 100 comprises aninput unit 110, a memory 120, a processor 130 and an output unit 140.The input unit 110 receives input data and user commands from anysuitable interface. The memory 120 and the processor 130 together form aprocessor unit 150 that operates to perform calculations on the inputdata and to store the results of these calculations in the memory 120.The output unit 140 outputs the stored results through any suitableinterface onto any suitable recording medium or display device. In oneexample, the apparatus 100 is a general purpose computing platform, suchas a desktop computer or server computer, as will be familiar to thoseskilled in the art.

As shown in FIG. 1, the memory 120 and the processor 130 provide a tool160 that performs financial modelling according to a predetermined model170. This tool 160 uses the input data to perform calculations accordingto the predetermined model 170, and produces one or more outputsrepresenting data relevant to the set of financial assets.

In one example, the input unit 110 is arranged to receive signalscontaining data having information relating to a set of assets.Conveniently, the input unit 110 receives price information relating toa plurality of financial assets. In one example, the input unit 110receives an asset identifier for each asset and at least one price orreturn relating to each asset. Optionally, the price information is moredetailed and may include further variables, such as a historical priceset giving price information at various points in time relating to thisasset. The price information may be incomplete, e.g. for some assets theprice is provided for some points in time but not others. Optionally,the received price data is then processed by the tool 160 to provide aworking set of prices, such as by interpolation to fill in the missingdata items. In one example, the input unit 110 is arranged to receivesignals containing data having risk information relating to the set ofassets. As an example, the risk information is option impliedvolatility. In one example, the input unit 110 receives impliedvolatility data without reference to prices (or returns). In oneexample, the input unit 110 receives both the price information and therisk information.

The output unit 140 suitably records the output data onto anon-transient computer-readable recording medium. In one example, theoutput unit 140 outputs the data onto a hard disk drive or an opticaldisc. In another example, the output unit 140 further outputs the dataonto a human-readable display device, such as a printer or a displayscreen.

The present model 170 is based on an improved description of the riskassociated with the assets. When the primary preference of investors infinancial assets is for greatest financial return for the lowestpossible risk over an investment window, and if all rational investorsbehave in this manner, then prices should move to reflect thispreference. When risk can be attributed to any asset to have a positivemagnitude, it is possible to describe this risk as a vector whose basisis the set of all orthogonal, and hence independent, risk factors. Tworisky assets have risks which are independent when the scalar product oftwo vectors describing the risk of each asset is zero. In the presentmodel, an equilibrium occurs between two or more assets or portfolioswhich are described by independent orthogonal risk basis vectors whenthe excess expected return over cash divided by the risk magnitude isidentical for all of the individual risk basis vectors. To investoptimally in terms of the excess expected return to risk ratio when suchan equilibrium has been established between the assets in an investmentuniverse requires a mechanism for providing such an optimal portfoliosubject to operational constraints. The present model 170 described hereprovides the weightings of the assets for such an optimal investmentportfolio.

The present model 170 identifies that the pricing of independent risk inefficient financial markets must be such that an equilibrium isestablished. Here, the ratio of excess expected return to risk for twodifferent independent sources of risk should be the same. In otherwords, independent risks are rewarded commensurately.

The present model 170 describes the risk of a general asset as a vectorin the space of independent risks. The magnitude of the vector is thescalar measure of risk. A Risk Relation Matrix is made up of elementswhich are the scalar products of the risk vectors of pairs of asset riskvectors, represented in a basis of independent risk vectors. Risk is aninput to the system. Asset risk implicitly reflects all availableinformation that affects price; such information may include, forexample, the supply of and demand for any financial asset or set ofassets.

Since the excess expected return to risk ratio is the same for allindependent risks describing risk of the assets in the model at anygiven time, and given the description of risk within the model, theexpected excess returns of an asset is determined within the model as ascalar multiple of the sum of the expected excess returns (also known asrisk premia) of all the independent risks represented in an asset riskvector. This provides a risk premium (expected excess return) to withinthe same scalar multiple for each asset in the investment universe. Therelative expected excess returns can then be used in an objectivefunction. The objective function can then be maximised subject toconstraints to produce the efficient portfolio. The portfolio weightsprovided by the model are optimal weights when independent risks arepriced according to the model. The model objective function alsoprovides a unique measure of diversification.

FIG. 2 is a schematic flowchart of processes performed within theexample apparatus of FIG. 1. Here, the flowchart illustrates variousinputs that are received by the processes A-G, and the outputs that maybe generated from these processes.

Process A: From the set of all possible assets available to investors, asubset of these is selected. The selection may be produced by screeningthe set of all possible investments by any suitable criterion; in someembodiments the screen can be based upon liquidity of the asset; in someembodiments the screen can be based upon the market capitalization ofthe asset; in some embodiments the screen can be activated on an assetbecause of a lack of sufficient data to proceed with the method; anyscreening criterion that leaves assets in the selected universe can beused within the system. The Input for Process A is the set of all assetsavailable to investors and screens employed. The size of the assetuniverse selected is denoted here as N.

Process B: Data is collected for the set of all assets in the universeof assets selected in Process A. In some embodiments this data includesasset identifiers; in some embodiments this data includes asset priceson a periodic frequency; in some embodiments this data includes expectedrisks over an investment period; in some embodiments this data includesexpected risk correlation between assets over an investment period; insome embodiments this data includes the Risk Relation Matrix between theassets; in some embodiments the risk is given by asset volatilitiesimplied from derivative asset prices over a known tenor (time basis).

Process C: A Risk Relation Matrix expected over the investment period isrequired by the process. It is denoted here by the symbol v. In someembodiments, this is provided as input to the system; in someembodiments, the Risk Relation Matrix is determined from the input dataprovided to Process B. In some embodiments, the Risk Relation Matrix canbe approximated by a Variance-Covariance matrix; in some embodiments theRisk Relation Matrix is determined from the asset risk data; in someembodiments it is determined from the risk correlation matrix; in someembodiments the Risk Relation Matrix is symmetric positive definite; insome embodiments the output of Process C is the Risk Relation Matrix.

Process D: The Independent Risk Portfolios are provided by Process D; insome embodiments the Independent Risks are determined from theeigenvector matrix, E, and the corresponding diagonal eigenvalue matrix,Λ, of the Risk Relation Matrix, v, so that

V=E·Λ·E′  (1)

In this equation, the symbol “.” represents the operation of an innerproduct between matrices. In some embodiments one or more eigenvectorsof v, each a column of the matrix E, have all elements multiplied bynegative 1 to ensure that the risk premium of the correspondingindependent risk vector is positive. In some embodiments the componentsof the independent risk vectors are given as the components of theeigenvector scaled by the positive square root of the associatedeigenvalue so that the magnitude of each independent risk vector is oneby construction. In some embodiments the risk of each asset in theselected universe is expressed as a vector in the basis of theindependent risks, where the magnitude of the said vector is the risk ofthe said asset given by the norm of the said vector. In some embodimentsthe components of the asset risk vector in the basis of independentrisks for an asset is given by the row of the matrix product E·Λ^(1/2)corresponding to the same said asset. In some embodiments some of theindependent risk vectors can be neglected so that a subspace of thebasis of the independent risk vectors is employed by the process. TheProcess D provides the components of the risk vector for each asset inthe basis of the independent risk vectors, or any other appropriatederived basis.

Process E: The Expected Excess Returns over cash, denoted here as thevector R, to within a common scalar multiple are represented in someembodiments as the sum of the components of the vector of independentrisks so that, given as scalar product of the eigenvector matrix E, thesquare root of the eigenvalue matrix Λ^(1/2) and a column vector all ofwhose elements have value one, denoted by 1,

R∝E·Λ^(1/2)·1  (2)

Process F: The objective function D given in some embodiments of themodel by the expression:

$\begin{matrix}{D = \frac{w^{\prime} \cdot E \cdot \Lambda^{1/2} \cdot 1}{\sqrt{w^{\prime} \cdot V \cdot w}}} & (3)\end{matrix}$

where w represents a column vector containing portfolio weights, E isthe eigenvalue matrix and Λ is the corresponding diagonal eigenvaluematrix of the Risk Relation Matrix V for the universe of assetsselected. The objective function D may be represented in otherequivalent forms.

In some embodiments, operational constraints on the magnitude of theasset weights are provided to the system. In some embodimentsoperational derived quantities derived from the weights of the assetsare provided to the system. In some embodiments the constraints includea maximum total exposure in terms of the sum of asset weights. In someembodiments constraints derived from asset properties are employed. Theconstraints derived from asset properties are possibly non-linear. Theconstraints are, for example, total portfolio risk, or tracking error toanother portfolio. In some embodiments the weights may be constrained tobeing positive.

In one embodiment, the objective function D is maximised by variation ofthe asset weights w subject to the operational constraints provided tothe system. The process need not be constrained in some embodiments toprovide the resulting portfolio of weights w to within a scaling factoras

w∝E·Λ^(−1/2)·1  (4)

The resulting weights w from any embodiment are then provided as avector of real numbers as output from Process F.

Process G: Any portfolio weightings for the set of assets selected inProcess A can be provided to Process G. In some embodiments theportfolio weights w are provided to Process G from Process F. In someembodiments the portfolio weights w are provided directly as input toProcess G. In the Process G, the objective function D is determined as

$\begin{matrix}{D = \frac{w^{\prime} \cdot E \cdot \Lambda^{1/2} \cdot 1}{\sqrt{w^{\prime} \cdot V \cdot w}}} & (5)\end{matrix}$

and is provided as a scalar diversification number (or equivalentlymeasure) as output from Process G.

Process H: This process collects the weights determined from process Fapplied to data related to calendar date/times at a given frequency. Insome embodiments this will be monthly; in some embodiments thisfrequency will be quarterly; in some embodiments it will be at someother frequency. The asset price series input and asset weights providedby the Process F on different dates are used to create a Price Index,possibly adjusted for corporate actions like dividends. In one example,the Price Index is output by the model as a Benchmark Index.

In summary, the apparatus and method described herein provide a novelportfolio construction that can improve the risk adjusted return of aset of assets for asset prices, risks and expected returns relative tothe other portfolio construction methods. In some example embodiments,the apparatus and method provide: a risk relation matrix; AssetIndependent Risk vectors; optimal constrained and unconstrainedportfolio weights with the best expected returns; a diversificationmeasure for portfolios; and/or a benchmark index that can be employed byinvestors against which they can measure the performance of theirinvestments. When asset markets are in a desired equilibrium, theportfolio formed by the weights provided by the system and method canhave the best expected excess returns for unit risk.

At least some of the example embodiments may be constructed, partiallyor wholly, using dedicated special-purpose hardware. Terms such as‘component’, ‘module’ or ‘unit’ used herein may include, but are notlimited to, a hardware device, such as a Field Programmable Gate Array(FPGA) or Application Specific Integrated Circuit (ASIC), which performscertain tasks.

Also, elements of the example embodiments may be configured to reside onan addressable storage medium and be configured to execute on one ormore processors. That is, some of the example embodiments may beimplemented in the form of a computer-readable storage medium havingrecorded thereon instructions that are, in use, executed by a computersystem. The medium may take any suitable form but examples includesolid-state memory devices (ROM, RAM, EPROM, EEPROM, etc.), opticaldiscs (e.g. Compact Discs, DVDs, Blu-Ray discs and others), magneticdiscs, magnetic tapes and magneto-optic storage devices.

In some cases the medium is distributed over a plurality of separatecomputing devices that are coupled by a suitable communications network,such as a wired network or wireless network. Thus, functional elementsof the invention may in some embodiments include, by way of example,components such as software components, object-oriented softwarecomponents, class components and task components, processes, functions,attributes, procedures, subroutines, segments of program code, drivers,firmware, microcode, circuitry, data, databases, data structures,tables, arrays, and variables.

Further, although the example embodiments have been described withreference to the components, modules and units discussed herein, suchfunctional elements may be combined into fewer elements or separatedinto additional elements.

Although a few preferred embodiments have been shown and described, itwill be appreciated by those skilled in the art that various changes andmodifications might be made without departing from the scope of theinvention, as defined in the appended claims.

Attention is directed to all papers and documents which are filedconcurrently with or previous to this specification in connection withthis application and which are open to public inspection with thisspecification, and the contents of all such papers and documents areincorporated herein by reference.

All of the features disclosed in this specification (including anyaccompanying claims, abstract and drawings), and/or all of the steps ofany method or process so disclosed, may be combined in any combination,except combinations where at least some of such features and/or stepsare mutually exclusive.

Each feature disclosed in this specification (including any accompanyingclaims, abstract and drawings) may be replaced by alternative featuresserving the same, equivalent or similar purpose, unless expressly statedotherwise. Thus, unless expressly stated otherwise, each featuredisclosed is one example only of a generic series of equivalent orsimilar features.

The invention is not restricted to the details of the foregoingembodiment(s). The invention extends to any novel one, or any novelcombination, of the features disclosed in this specification (includingany accompanying claims, abstract and drawings) or to any novel one, orany novel combination, of the steps of any method or process sodisclosed.

1. A numerical modelling apparatus, the apparatus comprising: an inputunit arranged to receive signals containing data having informationrelating to a set of assets; a processor unit arranged to: a) provide aRisk Relation Matrix v having a plurality of elements, wherein each ofthe elements represents a relationship of risk related to a respectivepair of the assets and each element is given by a scalar product of tworisk vectors, such that each of the assets has an associated risk vectoraccording to the elements of the risk relation matrix; b) decompose theRisk Relation Matrix v into eigenvectors and eigenvalues according to:V=E·Λ·E′ wherein E is a set of eigenvectors of the risk matrix v incolumns, Λ is the corresponding diagonal eigenvalue matrix, and E′ isthe transpose of E; and c) derive components of each of the risk vectorsin the basis of unit independent risks by the corresponding row of thematrix product E·Λ^(1/2) relating to each of the assets; and an outputunit arranged to output the components of each of the risk vectors as arisk vector data set onto a tangible computer-readable recording mediumor a display device.
 2. The apparatus of claim 1, wherein the input unitis arranged to receive risk information relating to the set of assetsand provide the risk information to the processor unit.
 3. The apparatusof claim 2, wherein the input unit is arranged to receive the RiskRelation Matrix as the risk information and to provide the Risk RelationMatrix to the processor unit.
 4. The apparatus of claim 2, wherein theinput unit is arranged to receive the risk vectors as the riskinformation and the processor unit is arranged to provide the RiskRelation Matrix from the risk vectors.
 5. The apparatus of claim 1,wherein the processor unit is arranged to provide a dataset of portfolioweights w for each of the assets according to:w∝E·Λ^(−1/2)·1 where E is a set of eigenvectors of the risk matrix v incolumns and Λ is the diagonal eigenvalue matrix and 1 is a column vectorall of whose elements are equal to one.
 6. The apparatus of claim 5,wherein the output unit is arranged to output the portfolio weightings wrelating to the set of assets.
 7. The apparatus of claim 1, wherein theprocessor unit is arranged to provide a dataset of expected returns Rfor each of the assets as a vector according to:R∝E·Λ^(1/2)·1 where E is a set of eigenvectors of the risk matrix v incolumns and Λ is the corresponding diagonal eigenvalue matrix and 1 is acolumn vector all of whose elements are equal to one.
 8. The apparatusof claim 7, wherein the output unit is arranged to output the expectedreturns R relating to the set of assets.
 9. The apparatus of claim 1,wherein the processor unit is arranged to obtain a dataset of portfolioweightings w relating to each of the assets, wherein the portfolioweightings w are provided as a column vector from:$D = \frac{w^{\prime} \cdot E \cdot \Lambda^{1/2} \cdot 1}{\sqrt{w^{\prime} \cdot V \cdot w}}$where w represents a column vector containing portfolio weights, E isthe eigenvector matrix and Λ is a corresponding diagonal eigenvaluematrix of the Risk Relation Matrix V for the set of assets, and D is anObjective Function relating to the set of assets.
 10. The apparatus ofclaim 9, wherein the output unit is arranged to output the value of theObjective Function as a Diversification Measure relating to the set ofassets.
 11. The apparatus of claim 9, wherein the input unit is arrangedto receive one or more constraints relating to the set of assets, andthe processor unit is arranged to optimise the Objective Function D byvariation of the portfolio weights w subject to the constraints.
 12. Theapparatus of claim 9, wherein the processor unit is further arranged tocollect the portfolio weights w at a plurality of time intervals. 13.The apparatus of claim 12, wherein the processor unit is furtherarranged to provide a Price Index based on the price information and thecollected weights relating to the set of assets.
 14. The apparatus ofclaim 13, wherein the output unit is arranged to output the Price Indexas a Benchmark Index relating to the set of assets.
 15. A method ofperforming numerical modelling using a computer apparatus, the methodcomprising: receiving signals at an input unit of the computerapparatus, the signals containing data having information relating to aset of assets; providing a Risk Relation Matrix v using a processor unitof the computer apparatus, wherein the Risk Relation Matrix v comprisesa plurality of elements, wherein each of the elements represents arelationship of risk related to a respective pair of the assets and eachelement is given by a scalar product of two risk vectors, such that eachof the assets has an associated risk vector according to the elements ofthe risk relation matrix; decomposing the Risk Relation Matrix v intoeigenvectors and eigenvalues according to:V=E·Λ·E′ wherein E is a set of eigenvectors of the risk matrix v incolumns, Λ is the corresponding diagonal eigenvalue matrix, and E′ isthe transpose of E; and deriving components of each of the risk vectorsin the basis of unit independent risks by the corresponding row of thematrix product E·Λ^(1/2) relating to each of the assets; and outputtingthe components of each of the risk vectors as a risk vector data setonto a tangible computer-readable recording medium or a display device.16. A tangible computer-readable recording medium having recordedthereon instructions which, upon execution by a computer apparatus,cause the performance of operations comprising: receiving signals at aninput unit of the computer apparatus, the signals containing data havinginformation relating to a set of assets; providing a Risk RelationMatrix v using a processor unit of the computer apparatus, wherein theRisk Relation Matrix v comprises a plurality of elements, wherein eachof the elements represents a relationship of risk related to arespective pair of the assets and each element is given by a scalarproduct of two risk vectors, such that each of the assets has anassociated risk vector according to the elements of the risk relationmatrix; decomposing the Risk Relation Matrix v into eigenvectors andeigenvalues according to:V=E·ΛE′ wherein E is a set of eigenvectors of the risk matrix v incolumns, Λ is the corresponding diagonal eigenvalue matrix, and E′ isthe transpose of E; and deriving components of each of the risk vectorsin the basis of unit independent risks by the corresponding row of thematrix product E·Λ^(1/2) relating to each of the assets; and outputtingthe components of each of the risk vectors as a risk vector data setonto a tangible computer-readable recording medium or a display device.